wu :: forums « wu :: forums - Interesting inequality » Welcome, Guest. Please Login or Register. May 25th, 2024, 8:27am RIDDLES SITE WRITE MATH! Home Help Search Members Login Register
 wu :: forums    riddles    putnam exam (pure math) (Moderators: SMQ, Icarus, towr, Grimbal, Eigenray, william wu)    Interesting inequality « Previous topic | Next topic »
 Pages: 1 Reply Notify of replies Send Topic Print
 Author Topic: Interesting inequality  (Read 1230 times)
wonderful
Full Member

Posts: 203
 Interesting inequality   « on: Jun 30th, 2008, 8:18pm » Quote Modify

Can you generalize the result?

Have A Great Day!
 « Last Edit: Jul 1st, 2008, 1:25pm by wonderful » IP Logged
towr
wu::riddles Moderator
Uberpuzzler

Some people are average, some are just mean.

Gender:
Posts: 13730
 Re: Interesting inequality   « Reply #1 on: Jul 1st, 2008, 12:54am » Quote Modify

Shouldn't the second term have z2+2xz in the numerator ?
 IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
ThudnBlunder
Uberpuzzler

The dewdrop slides into the shining Sea

Gender:
Posts: 4489
 Re: Interesting inequality   « Reply #2 on: Jul 1st, 2008, 5:54am » Quote Modify

on Jul 1st, 2008, 12:54am, towr wrote:
 Shouldn't the second term have z2+2xz in the numerator ?

Undoubtedly. If so, I get f(x,y,z) > 3, but what do I know?
 IP Logged

THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
pex
Uberpuzzler

Gender:
Posts: 880
 Re: Interesting inequality   « Reply #3 on: Jul 1st, 2008, 6:55am » Quote Modify

on Jul 1st, 2008, 5:54am, ThudanBlunder wrote:
 Undoubtedly. If so, I get f(x,y,z) > 3, but what do I know?

If we write f(x, y, z) = (y2 + 2yz) / (y - z)2 + (z2 + 2xz) / (x - z)2 + (x2 + 2xy) / (x - y)2,
then isn't lim(m -> infinity) f(1, m, m2) = 1?
 IP Logged
towr
wu::riddles Moderator
Uberpuzzler

Some people are average, some are just mean.

Gender:
Posts: 13730
 Re: Interesting inequality   « Reply #4 on: Jul 1st, 2008, 7:07am » Quote Modify

on Jul 1st, 2008, 6:55am, pex wrote:
 isn't lim(m -> infinity) f(1, m, m2) = 1?
That's what I was gonna say (well almost).

Maybe we're supposed to assume x,y,z are integers?
 IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
pex
Uberpuzzler

Gender:
Posts: 880
 Re: Interesting inequality   « Reply #5 on: Jul 1st, 2008, 7:09am » Quote Modify

on Jul 1st, 2008, 7:07am, towr wrote:
 Maybe we're supposed to assume x,y,z are integers?

Wouldn't the same counterexample still work?

Edit: I realize you probably derived the counterexample the same way I did, using (1/m, 1, m). However, f is homogeneous of degree zero...
 « Last Edit: Jul 1st, 2008, 7:10am by pex » IP Logged
towr
wu::riddles Moderator
Uberpuzzler

Some people are average, some are just mean.

Gender:
Posts: 13730
 Re: Interesting inequality   « Reply #6 on: Jul 1st, 2008, 7:19am » Quote Modify

on Jul 1st, 2008, 7:09am, pex wrote:
 Wouldn't the same counterexample still work?
Err, ahum, yes..

Quote:
 Edit: I realize you probably derived the counterexample the same way I did, using (1/m, 1, m). However, f is homogeneous of degree zero...
Actually, I used f(1/m2, 1/m, 1)
Or rather, I picked two of them to be practically 0 (but of a different order)
 IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
ThudnBlunder
Uberpuzzler

The dewdrop slides into the shining Sea

Gender:
Posts: 4489
 Re: Interesting inequality   « Reply #7 on: Jul 1st, 2008, 9:41am » Quote Modify

Can't we also say as n -> 0, m -> infinity then f(n, m, m2) -> 0?
And as m -> infinity, f(m, m+1, z) -> infinity.
Hence the expression can take all positive values.

 IP Logged

THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
wonderful
Full Member

Posts: 203
 Re: Interesting inequality   « Reply #8 on: Jul 1st, 2008, 1:27pm » Quote Modify

Thanks so much guys for pointing out some typos in the original question. I have revised accordingly. FYI, here is the revised one:

Have A Great Day!
 IP Logged
wonderful
Full Member

Posts: 203
 Re: Interesting inequality   « Reply #9 on: Jul 1st, 2008, 8:28pm » Quote Modify

Here is a more general version:

Have A Great Day!
 IP Logged
pex
Uberpuzzler

Gender:
Posts: 880
 Re: Interesting inequality   « Reply #10 on: Jul 3rd, 2008, 2:32pm » Quote Modify

 hidden: By symmetry, we lose no generality in assuming 0 < x < y < z. Additionally, by homogeneity, we may set z = 1.   What remains is a function of two variables x and y. I haven't explicitly checked it, but it looks like the function is everywhere increasing in x, so that the function approaches its infimum as x -> 0. By continuity, we may set x = 0 for the moment to solve for y.   The remaining function of one variable can be differentiated. After simplifying, we need to find the roots of a seventh-degree polynomial. Three of them are easy to locate (one is -1 and the others are the complex roots of x2 - x + 1); we are left with a fourth-degree polynomial.   The roots of this polynomial can be found algebraically. The only one that lies between 0 and 1 is y = 3/4 + sqrt(5)/4 - sqrt(6*sqrt(5) - 2)/4.   We calculate f(0, 3/4 + sqrt(5)/4 - sqrt(6*sqrt(5) - 2)/4, 1) = 5/2 + 5*sqrt(5)/2.

Thus, the greatest lower bound is k = 5/2 + 5*sqrt(5)/2, attained when x -> 0, y = 3/4 + sqrt(5)/4 - sqrt(6*sqrt(5) - 2)/4, and z = 1. We observe that k is approximately equal to 8.0902 > 4.
 IP Logged
wonderful
Full Member

Posts: 203
 Re: Interesting inequality   « Reply #11 on: Jul 3rd, 2008, 5:09pm » Quote Modify

Well-done Pex! You arrive at the correct conclusion. Can you find a simpler solution? More particularly, can you find a way to come up with a simpler maximization programming?

Have A Great Day!
 IP Logged
wonderful
Full Member

Posts: 203
 Re: Interesting inequality   « Reply #12 on: Jul 4th, 2008, 2:26pm » Quote Modify

Hi Pex,

I looked at your solution and really like some the maximization techniques you used in the proof. Thanks for sharing.

Have A Great Day!

P.S. There are other solutions. If anyone find out, please feel free to post here.
 IP Logged
 Pages: 1 Reply Notify of replies Send Topic Print

 Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »